Balanced notation for $\tau$-rigid pairs

Abstract

We introduce a new notation for $\tau$-rigid pairs called ``balanced pairs''. The notation $\frac{A}{B}$ allows for simplification of some formulas, for example the Jasso category and its dual. Using this balanced notation we show that the $g$-vector $g(\frac{A}{B})$ has nice geometric and algebraic implications: It defines ``lower'' and ``upper'' chambers and we prove that this corresponds to generalized Bongartz and co-Bongartz completions of balanced pairs. We also show that the balanced notation agrees the wall labels for the semi-invariant picture in the hereditary case and, in the general case, we describe the relation between balanced notation and the wall labels.

37 pages, 10 figures. The idea behind this paper was presented at the 2021 Workshop on Cluster Algebras and Related Topics at the Morningside Center for Mathematics at CAS in Beijing, August 2021. v2: added details and theorems. Submitted to Axioms Journal